One-point height fluctuations and two-point correlators of $(2+1)$ cylindrical KPZ systems

Abstract

While the 1-point height distributions (HDs) and 2-point covariances of $(2+1)$ KPZ systems have been investigated in several recent works for flat and spherical geometries, for the cylindrical one the HD was analyzed for few models and nothing is known about the spatial and temporal covariances. Here, we report results for these quantities, obtained from extensive numerical simulations of discrete KPZ models, for three different setups yielding cylindrical growth. Beyond demonstrating the universality of the HD and covariances, our results reveal other interesting features of this geometry. For example, the spatial covariances measured along the longitudinal and azimuthal directions are different, with the former being quite similar to the curve for flat $(2+1)$ KPZ systems, while the latter resembles the Airy$_2$ covariance of circular $(1+1)$ KPZ interfaces. We also argue (and present numerical evidence) that, in general, the rescaled temporal covariance $\mathcal{A}(t/t_0)$ decays asymptotically as $\mathcal{A}(x) \sim x^{-\bar{\lambda}}$ with an exponent $\bar{\lambda} = \beta + d^*/z$, where $d^*$ is the number of interface sides kept fixed during the growth (being $d^* = 1$ for the systems analyzed here). Overall, these results complete the picture of the main statistics for the $(2+1)$ KPZ class.